Jaron's World

Jaron's World: Shapes in Other Dimensions | Physics & Math | DISCOVER Magazine

In this 3-D slice of the four-dimensional hendecatope, colored beams represent the
edges of triangles; some triangles are left out for simplicity.

(Computer model courtesy of Carlo Sequin, UC Berkeley, styled by Jaron Lanier

discovermagazine.com/2007/apr

These are shapes, like the cube and the tetrahedron (the regular three-sided pyramid) in which every angle, every facet, and every edge is identical. There are only five such shapes in the three-dimensional world; the other three are the octahedron (eight triangular sides), icosahedron (20 triangles), and dodecahedron (12 pentagons).

This was proved in ancient times by Euclid, and it is hard to overstate how profoundly amazing this proof must have been—and remains. The identities of the five shapes, and the certainty that there can be no more than five, is absolute and universal. While it is possible that an alien would never think to ask the question, all life everywhere would indisputably agree on the answer.

A mathematical proof is something anyone can do, yet it is bigger than the universe.